Schrödinger flow near harmonic maps

Schrödinger flow near harmonic maps

For the Schrödinger flow from R2 × R+ to the 2-sphere S2, it is not known if finite energy solutions can blow up in finite time. We study equivariant solutions whose energy is near the energy of the family of equivariant harmonic maps. We prove that such solutions remain close to the harmonic maps until the blowup time (if any), and that they blow up if and only if the length scale of the neare...

A pr 2 00 5 Schrödinger Flow Near Harmonic Maps Stephen Gustafson

Abstract For the Schrödinger flow from R × R to the 2-sphere S, it is not known if finite energy solutions can blow up in finite time. We study equivariant solutions whose energy is near the energy of the family of equivariant harmonic maps. We prove that such solutions remain close to the harmonic maps until the blow up time (if any), and that they blow up if and only if the length scale of th...

Se p 20 06 Asymptotic stability of harmonic maps under the Schrödinger flow ∗

For Schrödinger maps from R ×R+ to the 2-sphere S, it is not known if finite energy solutions can form singularities (“blowup”) in finite time. We consider equivariant solutions with energy near the energy of the two-parameter family of equivariant harmonic maps. We prove that if the topological degree of the map is at least four, blowup does not occur, and global solutions converge (in a dispe...

Asymptotic Stability , Concentration , and Oscillation in Harmonic Map Heat - Flow , Landau - Lifshitz , and Schrödinger Maps on R 2

We consider the Landau-Lifshitz equations of ferromagnetism (including the harmonic map heat-flow and Schrödinger flow as special cases) for degree m equivariant maps from R2 to S2. If m ≥ 3, we prove that near-minimal energy solutions converge to a harmonic map as t → ∞ (asymptotic stability), extending previous work (Gustafson et al., Duke Math J 145(3), 537–583, 2008) down to degree m = 3. D...

Stable Exponentially Harmonic Maps Between Finsler Manifolds